Learning

1-Statement of the Issue: Three reforms have been occurred in the USA to improve teaching and learning mathematics in schools in the twenty century. Based on these renew of mathematics education programs, students education also has changed. Reform under the banner of NCTM began in the recent decade. The aim of this movement is developing student’s mathematical power (Kilpatrick1992). This is the key word of the last reform. Most of researchers have found that students are coming to school with a lot of mathematical knowledge (Ginsburg &al 1997). D’ Ambrosio (1984,1985) has used the expression (ethno mathematics) to refer to forms of mathematics that vary as a consequence of being embedded in cultural activities. Thy have learned this knowledge of mathematics out of school and during their everyday communication and experience. Therefore early experiences establish a base for further mathematical knowledge that grows in school settings. Over the last decade scholars have revised considerably their assumption about mathematics learning. In mathematics classes teachers are as learner. Ann L. Brown mentioned in community of learning project the teacher is not far away from learning opportunity. The teacher learns along with the children as well as assists their efforts. She also believes that learners develop at different rates. So in a community as classroom, teacher would learn in his/her own rate. Most of scientists and philosophers and religious leaders have emphasized a long life learning for human beings. Over all academic learning is active, strategic, self-conscious, self-motivated and purposeful.

2-Brief review of relevant research: Today we have a large body of research findings on learning in cognitive domain. This knowledge is growing faster and is supported by educators, educational psychologists and teachers. Researchers and educationist psychologists revised their assumptions about human learning in the last decade based on research findings (Peterson 1994). Many of their ideas are not new and have evolved from ideas of Piaget, Dewey and vygotsky. Interviewing children and studying the development of children’s mathematical knowledge emerged that; students come to school with a lot of mathematical knowledge. From these studies of children mathematical knowledge, researchers developed framework of children’s mathematical knowledge in several mathematical domains (Carpenter 1986 and Riley & Greeno, 1988). So children try to understand and think about the new schooling knowledge in terms of they already know (Glaser. 1984). Recently researchers argue that mathematics learning is an individual, creative activity and a communal, social practice (Cobb, Yackel, and Wood, 1992).

Similarly researchers assume that learners have to construct their own knowledge-individually and collectively (Davis, Moher and Noddings 1990). If learning is viewed as social and communal as well as individual and constructive, then the question of how meaning comes to be shared becomes important. Therefore researchers are increasingly focusing on the potential conversational patterns of discourse as compared with the lecture recitation pattern of discourse that has traditionally predominated in classrooms (Cazden 1986). Constructivist, which has its roots in the ideas of Jean Piaget, takes point of view that individuals actively construct the mathematical knowledge in mathematics learning processes (from what they know to what they do not know). This construction of the knowledge is lifelong, effortful process requiring significant mental engagement by the learner individually and socially. So the new mathematical knowledge will not make sense to us and may be constructed in a way that is not useful for long term recall or for application in a variety of situations (Resnick 1983). Researchers by interview techniques concluded that children do invent solutions to mathematics problem (davis, 1984). Student’s literature is a body of Mathematical knowledge that emerged from research studies. This knowledge of mathematics in early childhood can help children to assimilate school knowledge of mathematics (Ginsburg, 1984 and NCTM, 1991).

Organizing mathematical knowledge is another issue; researchers have modeled in the recent years. Piaget model of learning organization is a structure of knowledge that, he nominates cognitive structure. These structures can be envisioned as constituting an interconnected hierarchical network. The fundamental principles and concepts occupied higher levels of hierarchy, and supplementary concepts and many examples and non-examples stored in the lowest levels (piaget & Inhelder 1965). Recently research findings suggest that experts and novices organize their mathematical knowledge in memory rather differently. Within such a model of knowledge, experts have more concepts, more relations about defining each part of knowledge more interrelation among concepts and finally effective methods for receiving related concepts (Mertre 1991; Glaser, 1992).

3-Major approaches: Three major views in mathematics learning have influenced. 1- Behaviorist view: In this view students to learn a complex process is to break down the process into component parts, learn each part then, put together the components to obtain desired behavior. 2- Levinien rationalist premise that human interaction is ultimately dependent on the cognitive processing of information that is on the world as cognized, not the world as it is. 3- Constructivist that represents the latest model of child-centered learning (Learman & Dengate1995). From the recent theory (constructivist view) an investigative approach has emerged that involves purposeful inquiry based on meaningful instruction and, thus, can foster all aspects of mathematical power (Baroody 1998). According to Piaget, logical knowledge consists of the cognitive structures that assimilate and accommodate to all incoming information (Piaget and Inhelder 1960).

4-Conclusions: Mathematics Learning Model: the model of learning is based on social constructivist views and The National Council of Teachers of Mathematics (NCTM) standards. The main goal of learning mathematics is fostering student’s mathematical power (NCTM 1989). Developing mathematical power can be interpreted in several ways. The important point is, students have to be able to apply their mathematical knowledge in other unfamiliar situations. Therefore the students should learn mathematics:

1- With willingness and the self-confidence.

2- To engage in process of inquiry by reasoning, conjecturing, solving challenging problems and communicating.

3- With deep understanding and connected to everyday life.

The roots of deep understanding of mathematics are in Jean Piaget’s cognitive Theory (Piaget & Inhelder 1965).

Learning model shows close relationship between learner and learning situation that teacher has planned before. Learner is dependent variable and learning situation is independent variable. When learner looses his/her equilibrium with learning situation, is going to obtain balance. On the way of obtaining balance, learner needs to try and shows up as an active learner in the learning scene. These processes continue until learner becomes satisfied. The products of these activities are learning, that assimilate or accommodate in the learner cognitive structure and then, transfer to long-term memory. In other word learning is producing new ideas and thoughts. In this model the duty of teacher is performing learner situation and leads learner to be active and productive in the learning situation.

Learner Touching learner needs motivation Learning activities

Interaction Y=f (x) Learning

x

Learning L.T Memory

Situation Role of the teacher(creating learning situation)

Learning Model

References:

Cazden.C.B. (1986). Classroom Discourse. In M.C. Wittrok (Ed.) Handbook of research on Teaching (Third edition) New York: Macmillan.

Cob. P. Yackel. E. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal of research in mathematics education.

Davis.R. (1984) Learning Mathematics. Norwood. NJ: Albex.

Davis.R.B. Maher. C. A.and Noddings.N. (Eds) (1990). Constructivist views on teaching and learning of mathematics. Journal for Research in Mathematics Education. Monograph 4. Reston. VA: National Council of Teachers of Mathematics.

Ginsberg, H, P., (1989). Children’s arithmetic: How they learn it and how you teach it (2^nd ed.). Austin, TX: Pro Ed.

Glaser. R. (1984). Education and Thinking: The role of knowledge. American psychology, 39,93-104.

Mestre Jose, Cognitive Aspects of Learning and Teaching Science. Teacher Enhancement for Elementary and Secondary science and Mathematics. (19940 Washington, D.C 20220

National Council of Teachers of Mathematics. (1991). Professional Standards for Teaching Mathematics.Reston.Va.

Peterson.L. Penelope. Learning and Teaching Mathematical Science: Implications for In-service Program. Teacher Enhancement for Elementary and Secondary science and Mathematics. (19940 Washington, D.C 20220.

Riley, M, S. & Greeno.j. G (1988). Developmental Analysis of Understanding language about quantities and Solving Problems. Cognition and Instruction.5, 49-101.